This project is intended to increase our understanding of the use and application of mathematical and statistical models in toxicology and biochemistry and to implement new mathematical models to aid in explaining current research findings. The research effort explores a diverse range of biological areas including carcinogenesis, pharmacology, developmental biology, neurology and immunology. In carcinogenesis; (1) an estimate of carcinogenic potency which adjusts for chemically related changes in survival was evaluated and a modification allowing for nonlinear dose-response was developed; (2) mathematical model of carcinogenesis which explicity incorporates DNA repair into the multistage process was applied to initiation-promotion experiments in the skin and liver models; (3) health risks from exposure to 2,3,7,8-TCDD are being explored via several mechanistically-based mathematical models; (4) mechanistic models for ligand-receptor binding were developed and their implications for risk assessment were explored; (5) the shape of the carcinogenesis dose-response curve using information on biological activity of the chemical has been studied and suggests that chemical structure correlates well with dose response shape; (6) a multistage model of carcinogenesis incorporating an explicit component for stem cells was developed and applied to an initiation-promotion study in mouse skin; (7) alternatives to the two- stage model of carcinogenesis utilizing multiple pathways to better incorporate the role of oncogenes are underway; (8) extra-Poisson variability in two-stage models of carcinogenesis is being explored. In teratology, it was found that resampling techniques and quasi-likelihood methods can be used to account for interlitter correlations when analyzing teratological data. In immunotoxicology, it was found that there is little or no relationship between immunotoxicity and mutagenicity but that carcinogenicity and immunotoxicity are related with immunotoxic compounds having a high probability of being carcinogenic. In neurology, a research effort is underway with NINDS to apply multivariate smoothing techniques to characterize the relationship between areas of the brain and different parts of the body. Also, the application of smoothing spline methods to the estimation of rate functions in pharmacology and biochemistry is underway. Finally, a graph theoretical modeling technique for qualitative determination of the regulatory properties of biochemical networks was developed.